An Economist Is in the Process of Developing a Model $\left( x _ { 1 } \right)$

Essay

An economist is in the process of developing a model to predict the price of gold. She believes that the two most important variables are the price of a barrel of oil $\left( x _ { 1 } \right)$ and the interest rate $$\left( x _ { 2 } \right)$$ She proposes the first-order model with interaction: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 3 } + \varepsilon$ . A random sample of 20 daily observations was taken. The computer output is shown below.
THE REGRESSION EQUATION IS
$$y = 115.6 + 22.3 x _ { 1 } + 14.7 x _ { 2 } - 1.36 x _ { 1 } x _ { 2 }$$
$\begin{array}{|c|rrc|}\hline \text { Predictor } & \text { Coef } & \text { S2Dev } & T \\\hline \text { Constant } & 115.6 & 78.1 & 1.480 \\x_{1} & 22.3 & 7.1 & 3.141 \\x_{2} & 14.7 & 6.3 & 2.333 \\x_{1} x_{2} & -1.36 & 0.52 & -2.615 \\\hline\end{array}$
$$\mathrm { S } = 20.9 \quad \mathrm { R } - \mathrm { Sq } = 55.4 \%$$
ANALYSIS OF VARIANCE
$\begin{array}{|l|rrrr|}\hline \text { Source of Variation } & d f & \text { SS } & \text { MS } & F \\\hline \text { Regression } & 3 & 8661 & 2887.0 & 6.626 \\\text { Error } & 16 & 6971 & 435.7 & \\\hline \text { Total } & 19 & 15632 & & \\\hline\end{array}$ Is there sufficient evidence at the 1% significance level to conclude that the interaction term should be retained?